Problem: If $x^2+y^2=1$, what is the largest possible value of $|x|+|y|$?
If $(x,y)$ lies on the circle, so does $(x,-y),$ $(-x,-y),$ and $(-x,-y),$ (which all give the same value of $|x| + |y|$), so we can assume that $x \ge 0$ and $y \ge 0.$

Then $|x| + |y| = x + y.$  Squaring, we get
\[(x + y)^2 = x^2 + 2xy + y^2 = 1 + 2xy.\]Note that $(x - y)^2 \ge 0.$  Expanding, we get $x^2 - 2xy + y^2 \ge 0,$ so $2xy \le x^2 + y^2 = 1.$  Hence,
\[1 + 2xy \le 2,\]which means $x + y \le \sqrt{2}.$  Equality occurs when $x = y = \frac{1}{\sqrt{2}},$ so the maximum value of $|x| + |y|$ is $\boxed{\sqrt{2}}.$